Notes on J-Holomorphic Maps

TL;DR

This paper systematically explores local properties and convergence behavior of J-holomorphic maps, clarifying assumptions and extending understanding within symplectic geometry, based on foundational chapters of McDuff-Salamon's work.

Contribution

It provides a detailed, assumption-specific analysis of J-holomorphic maps and their convergence, with minimal reliance on symplectic conditions, enhancing theoretical understanding.

Findings

Clarifies assumptions needed for properties of J-holomorphic maps

Details Gromov's convergence for sequences of such maps

Extends foundational results with minimal symplectic dependence

Abstract

These notes present a systematic treatment of local properties of J-holomorphic maps and of Gromov's convergence for sequences of such maps, specifying the assumptions needed for all statements. In particular, only one auxiliary statement depends on the manifold being symplectic. The content of these notes roughly corresponds to Chapters 2 and 4 of McDuff-Salamon's book on the subject.